Question

Find the domain of definition of the following function:
$f\left(x\right)={\log }_{2x-5}(x^2-3x-10).$

Answer 1

$f\left(x\right)={log}_{(2x-5)}(x^2-3x-10)$

For $f\left(x\right)$ to be real and defined

$x^2-3x-10>0$

$\Rightarrow (x-5)(x+2)>0$

Therefore, $x\in (-\infty ,-2)\cup (5,\infty )⟶\left(1\right)$

$2x-5>0$ and $2x-5\ne 1$

$x>\frac{5}{2}$ and $x\ne 3$

Therefore, $x\in (\frac{5}{2},3)\cup (3,\infty )⟶\left(2\right)$

From $1$ and $2$

Domain of $f\left(x\right)$ is $x\in (5,\infty ).$